Thermodynamics 7
Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it except for one or two small points. The third time you go through it, you know you don’t understand it, but by that time you are so accustomed to it that it doesn’t bother you anymore. -Arnold Sommerfeld
1. The First Law
The First Law of Thermodynamics, also known as the law of conservation of energy, states that energy can neither be created nor destroyed. In physics its use often involves the interconversion of kinetic energy and various forms of potential energy (e.g. chemical, gravitational or electrostatic energy). However in chemistry the first law is used with quantities of chemical concern such as the total energy of a system, the heat flow in or out of it, and the work done on or by the system. So the First Law is typically expressed as:
ΔE = q + w
The total energy of the system, E, is the sum of the kinetic and the potential energy. It is sometimes referred to as the internal energy, U. Of course the actual energy of a system cannot be measured. All that can be measured is the change in the energy, ΔE. The heat flow is referred to as "q," and the work done on or by the system is "w." While a different sign convention is sometimes used in physics or engineering, in chemistry the sign convention is always the same. Energy which flows into a system — as heat flow or work being done on the system — is positive; energy which flows out of a system is negative. In a chemical system, where work is pressure times the change in volume, the sign change can be thought of as follows: If work is done on the system, it will be squeezed into a smaller volume. Hence +work is associated with -ΔV. Similarly a system whose volume increases is doing work on the surroundings. Thus -w (which is what you have when the system does work on the surroundings) is associated with a +ΔV. This is summarized by:
w = - P ΔV
and therefore
ΔE = q - P ΔV
2. Calorimetry
The heat flow, q, cannot itself be measured. What we can measure is temperature change, which is a consequence of heat flow. The process of using temperature change to measure heat flow is known as calorimetry, and the device in which this is done is known as a calorimeter.
The fundamental equation of calorimetry is:
q = c m ΔT
Here q is the heat flow (usually the heat given off in a chemical reaction), m is the mass of the system, and ΔT is the change in the system's temperature. The specific heat, c, is the amount of heat required to change the temperature of a gram of substance by one degree. Specific heat was once measured in calories, the amount of heat required to raise the temperature of a gram of water by one degree Celsius. Thus the specific heat of water was one. Specific heats are now measured in joules, with the specific heat of water being 4.184 Joules/g × K.
In AP Chemistry, a calorimeter usually consists of water in a styrofoam coffee cup. A coffee cup is used because it insulates well and, thus, prevents heat from leaking into or out of the system. Styrofoam cups also have the advantage of being very light and therefore absorbing so little heat that they can be ignored in our calculations.
Suppose, for example, a chemical reaction takes place in a coffee cup calorimeter containing 60.0 g of water. The initial temperature of the water is 24.0° C and its final temperature is 27.4° C. How much heat is given off in the reaction?
q = c m ΔT
q = 4.184 J/g × deg ´ 60.0 g ´ 3.4 deg
q = 853 joules = 850 joules
Sometimes a larger calorimeter, heavy enough to affect the experiment, must be used. Rather than weighing the calorimeter and estimating its specific heat, the product of the weight and the specific heat is determined. This product, known as the heat capacity and abbreviated as "C," can then be used in calorimetric calculations. This can be stated as:
q = (cwater mwater + C) ΔT
The heat capacity of a calorimeter is usually determined once and then inscribed on the calorimeter.
The open calorimeter described above, in which the pressure remains constant, is only one way of measuring heat flow. Calorimetric calculations at constant pressure are written as:
q = cp m ΔT
Since this is what chemists normally do, the cp is usually written as just "c."
However, measurements can also be made at constant volume in a sealed calorimeter. A constant-volume calorimeter, usually referred to as a "bomb" calorimeter, often gives a different set of values. Calorimetry under conditions of constant volume is governed by the equation:
q = cv m ΔT
A reaction in a bomb calorimeter proceeds under conditions of constant volume. This means that no work is done by (or to) the system. Thus:
ΔE = q + w = q
Finally it should be said that processes which, like calorimetry, do not exchange heat with their surroundings are known as adiabatic processes.
3. Enthalpy
Chemical measurements are usually carried out in open containers, where the pressure is constant. If the reaction pulls air into the flask, work is being done on the system by the surroundings. If gas goes from the flask to the surroundings, work is being done by the system. We can still measure the amount of heat given off or taken in by the reaction, but it is no longer equal to the change in the internal energy of the system. Some of the heat is converted to work.
We get around this problem by introducing a term called enthalpy (H). Enthalpy is the sum of the system's internal energy plus the product of pressure and volume. Thus:
H = E + PV
The change in the enthalpy of a system is equal to the change in its internal energy plus the change in the product of the pressure times the volume of the system.
ΔH = ΔE + P ΔV
Combining this equation with the first law of thermodynamics, ΔE = q - P ΔV, we end with:
ΔH = q
That is, the change in enthalpy will be equal to the heat flow. This will prove useful later on.
4. Hess's Law
Hess's Law states that if a reaction occurs in steps, even if only theoretically, the enthalpy change associated with the reaction is equal to the sum of the enthalpy changes of the individual steps. The rules for using Hess's Law are as follows:
1) The equations corresponding to the steps must add up to the total reaction
2) Whatever is done to a step (e.g., doubling it) must be done to the ΔH for that step.
3) The ΔH values of the steps must add up to the ΔH of the reaction.
Let us, for example, use the enthalpies of combustion for carbon, hydrogen and ethanol to calculate the enthalpy of formation for ethanol. This corresponds to the reaction:
(1) 2 C + 3 H2 + ½ O2 ® C2H5OH
The three enthalpies of combustion correspond to the following reactions:
(2) C + O2 ® CO2 ΔH° = -394 kJ/mol
(3) H2 + ½ O2 ® H2O ΔH° = -286 kJ/mol
(4) C2H5OH + 3 O2 ® 2 CO2 + 3 H2O ΔH° = -1366 kJ/mol
To get the desired reaction, we add two times the enthalpy change of reaction 2, three times the ΔH of reaction 3 and subtract the ΔH of reaction 4. This gives us, for reaction 1, ΔH = -278 kJ/mol.
Hess's Law works because enthalpy is a state function. That is, a given substance under a given set of conditions will always have the same enthalpy. Enthalpy depends on the state of the substance, not on how the substance got to where it is. Other state functions include entropy and free energy, which we will discuss shortly, as well as internal energy, volume, temperature and pressure. Work and heat are not state functions. You can talk about the entropy, the internal energy or the volume of a system. You cannot talk about the work of a system or the heat of a system. It doesn't sound right.
5. Standard Enthalpies (Heats) of Formation
Since enthalpy is a state function, tables of enthalpy values for various substances have been compiled. The conditions at which these values are obtained, 25° C and unit concentration (1 atm or 1 M), are called standard conditions and these values are referred to as standard enthalpies. Actually the tabulated values are not enthalpies but are enthalpy changes, since enthalpy values can only be calculated relative to an arbitrary zero. The zero of enthalpy is defined as the enthalpy of an element in its standard state. So the enthalpy of water is the enthalpy change which occurs in this reaction:
H2(g) + ½ O2(g) ® H2O(g)
The enthalpy change is called the "standard enthalpy of formation" (or the standard heat of formation) and is abbreviated ΔH°f. Notice in the above equation that the standard heat of formation for the reactants is zero, since both are in their standard states.
To predict the enthalpy change for a reaction you add the ΔH°f values of the products and subtract those of the reactants. Remember: products minus reactants. Take for example the reaction:
C2H5OH(l) + 3 O2(g) ® 2 CO2(g) + 3 H2O(l)
The enthalpy change for this reaction can be calculated as:
ΔH° = ΔH°f (products) - ΔH°f (reactants)
ΔH° = 2 ´ ΔH°f (CO2) + 3 ´ ΔH°f (H2O) - 1 ´ ΔH°f (C2H5OH) - 3 ´ H°f (O2)
ΔH° = 2 ´(-393.5) + 3 ´ (-285.8) - 1 ´ (-277.1) - 3 ´ (0)
ΔH° = -1367.3 kJ/mol
6. Bond Energy
One way of calculating the enthalpy change in a reaction is from the bond energies. One adds the energy of the bonds in the products and subtracts the energy of the bonds in the reactants. So the enthalpy change is the energy of the broken bonds minus the energy of the bonds which are formed. Please remember "broken bonds minus formed bonds."
Let us consider, as an example, the reaction:
N2(g) + 3 H2(g) D 2 NH3(g)
The relevant bond energies are:
NºN 941 kJ/mol
H - H 432 kJ/mol
N - H 391 kJ/mol
The enthalpy change in the reaction is calculated as below. Notice the sign change.
ΔH = broken bonds - formed bonds
ΔH = (3 ´ EH-H) + (1 ´ EN-N) - (6 ´ EN-H)
ΔH = (3 ´ 432) + (1 ´ 941) - (6 ´ 391)
ΔH = -109 kJ/mol
7. Entropy
If the energy in the Universe remains constant, if energy cannot be consumed, then what drives chemical reactions? It is the dispersal of energy. Reactions proceed when they allow energy to spread out. This tendency of energy to disperse is measured by something called "entropy."
When a hot object cools, the heat energy which was concentrated in the object spreads to the molecules — air and table molecules, for example — which touch the hot object.
When you drop an object its potential energy is converted to kinetic energy which then changes — very quickly — to thermal energy when it hits the ground.
When a gas expands, the energy of the gas molecules is spread out more widely.
When ice melts, thermal energy from a (warm) external source spreads into the ice and breaks the bonds holding the water molecules in their crystal lattice.
Even a spontaneous, exothermic reaction, a case where entropy doesn't seem necessary, is still a case where energy is spread out. Heat is transferred from the bonds of reactant to thermal energy in the system and its surroundings.
The Second Law of Thermodynamics states that “in any spontaneous process, the entropy of the Universe increases.” All of the above processes are spontaneous (or irreversible) since energy is being dispersed. An example of a non-spontaneous (reversible) process would be an ice cube at 0° C in a glass of water at 0.0001° C. Energy is being (slowly) transferred, but the temperature difference is so small that the system is virtually at equilibrium.
The change in entropy caused by one of these processes is calculated by taking the heat which is transferred and dividing it by the temperature at which the transfer occurs. Thus:
This equation is Clausius' original, thermodynamic definition of entropy, proposed around 1850. You will often see it written with a negative sign! This is because in physics and engineering a positive heat flow is from the system to the surroundings.
What is the entropy change associated with melting 1 gram of ice? Melting ice requires that 6.01 kjoules/mol flow into the system. (This is known as the enthalpy or the heat of fusion.) Thus:
ΔS = ______
273 K
Notice the entropy is measured in joules per degree, not in kilojoules. This is not an obvious point.
8. The Statistical Definition of Entropy
A more recent (around 1900) definition of entropy is the statistical definition. This definition, which was put forth by Ludwig Boltzmann, gives the same results but in a different way. The statistical definition says that entropy is determined by the number of microstates — the number of different arrangements — in which a system can exist. The relevant equation is:
S = k ln W
Here k is a constant, known as the Boltzmann constant, which is equal to R divided by Avogadro's Number. W is the number of microstates — the ways in which a system can be arranged.